CERN-TH-2020-061

Towards a unified equation of state for multi-messenger astronomy

Micha l Marczenko,1, ∗ David Blaschke,1, 2, 3 Krzysztof Redlich,1, 4 and Chihiro Sasaki1

1Institute of Theoretical Physics, University of Wroclaw, PL-50204 Wroclaw, Poland2Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Russia

3National Research Nuclear University, 115409 Moscow, Russia4Theoretical Physics Department, CERN, CH-1211 Geneve 23, Switzerland

(Dated: September 17, 2020)

We aim to present a first step in developing a benchmark equation-of-state (EoS) model for multi-messenger astronomy that unifies the thermodynamics of quark and hadronic degrees of freedom.A Lagrangian approach to the thermodynamic potential of quark-meson-nucleon (QMN) matterwas used. In this approach, dynamical chiral-symmetry breaking is described by the scalar mean-field dynamics coupled to quarks and nucleons and their chiral partners, whereby its restorationoccurs in the hadronic phase by parity doubling, as well as in the quark phase. Quark confinementwas achieved by an auxiliary scalar field that parametrizes a dynamical infrared cut-off in the quarksector, serving as an ultraviolet cut-off for the nucleonic phase space. The gap equations were solvedfor the isospin-symmetric case, as well as for neutron star (NS) conditions. We also calculated themass-radius (MR) relation of NSs and their tidal deformability (TD) parameter. The obtained EoSis in accordance with nuclear matter properties at saturation density and with the flow constraintfrom heavy ion collision experiments. For isospin-asymmetric matter, a sequential occurrence oflight quark flavors is obtained, allowing for a mixed phase of chirally-symmetric nucleonic matterwith deconfined down quarks. The MR relations and TDs for compact stars fulfill the constraintsfrom the latest astrophysical observations for PSR J0740+6620, PSR J0030+0451, and the NSmerger GW170817, whereby the tension between the maximum mass and compactness constraintsrather uniquely fixes the model parameters. The model predicts the existence of stars with a coreof chirally restored but purely hadronic (confined) matter for masses beyond 1.8 M. Stars withpure-quark matter cores are found to be unstable against the gravitational collapse. This instabilityis shifted to even higher densities if repulsive interactions between quarks are included.

I. INTRODUCTION

The equation of state (EoS) is one of the key observ-ables characterizing properties of matter under extremeconditions. In the context of strongly interacting matter,the EoS encodes information on the phase structure andthe phase diagram of quantum chromodynamics (QCD).

At a finite temperature and low net-baryon density,the EoS, obtained in ab initio calculations within lat-tice QCD (LQCD), exhibits a smooth crossover fromhadronic matter to a quark-gluon plasma, which is linkedto the restoration of the chiral symmetry and color de-confinement [1, 2]. LQCD provides the state-of-the-artresults describing properties of the low baryon-densityQCD matter that are also successfully used for the inter-pretation of data obtained from the relativistic heavy ioncollisions (HIC) [3–8].

The chiral symmetry restoration is explicitly identi-fied in LQCD calculations through the scaling propertiesof the chiral condensate or the behavior of observablesthat are sensitive to chiral criticality, such as fluctua-tions of conserved charges [8]. Recently, the chiral sym-metry restoration was explored via the temperature de-pendence of parity doublers when approaching the QCDphase boundary. The LQCD findings [9–11] exhibit aclear manifestation of the parity doubling structure for

∗ michal.marczenko@uwr.edu.pl

the low-lying baryons around the chiral crossover. Themasses of the positive-parity ground states are found tobe rather temperature-independent, while the masses ofnegative-parity states drop substantially when approach-ing the chiral crossover temperature, Tc. The parity dou-blet states become almost degenerate with a finite massin the vicinity of the chiral crossover. Despite the factthat these LQCD results are still not obtained in thephysical limit, the observed behavior of parity partnersis likely an imprint of the chiral symmetry restorationin the baryonic sector of QCD. It is to be expected thatsimilar results can also appear in the cold and dense nu-clear matter when approaching the chiral symmetry re-stored phase. Indeed, such properties of the chiral part-ners can be described in the framework of the parity dou-blet model [12–14]. The model has been applied to hotand dense hadronic matter, neutron stars, as well as thevacuum phenomenology of QCD [15–30].

Until now, LQCD calculations at nonzero baryonchemical potential suffer from the sign problem, whichstymies their application to the region of low tempera-ture and high net-baryon density. This domain of theQCD phase diagram and the corresponding EoS are ofparticular importance for the understanding of the ex-traterrestrial experiments, particularly for the study ofcompact stellar objects, especially neutron stars (NSs),their mergers [31], and supernovae [32].

The progress in constraining the EoS at a low temper-ature and high density was driven mostly due to dis-coveries of various two-solar mass NSs [33–36]. The

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modern observatories for measuring masses and radiiof compact objects, the gravitational wave interfer-ometers of the LIGO/Virgo Collaboration (LVC), andthe X-ray observatory Neutron star Interior Compo-sition Explorer (NICER) provide new powerful con-straints on the neutron-star mass-radius profile. The firstever detection of gravitational waves from the compactstar merger GW170817 [37], the second detection fromGW190425 [38], as well as the NICER observation of themillisecond pulsar PSR J0030+451 [39, 40] delivered si-multaneous measurements of masses and radii.

Recent advances in nuclear theory have also tightenedthe constraints on the EoS over a wide range of densities.This has been achieved by systematic analyses of newastrophysical observations within simplistic approaches,such as the constant-speed-of-sound (CSS) model [41] ormultipolytropic class of EoSs [42–44]. Such schemes areinstructive but not microscopic approaches. Althoughthey provide interesting heuristic guidance, they cannotreplace a realistic model for the EoS.

One of the uncertainties when modeling the EoS ofdense nuclear matter is NS composition. Due to presentlylimited knowledge, their internal structure remains un-clear. Different studies indicate that a variety of exoticconstituents may appear in the core of NSs [45], such ashyperons or meson condensates (see, e.g., [46, 47]). How-ever, the 2 M constraint is hard to reconcile when ad-ditional degrees of freedom are taken into account, sinceit is known that additional degrees of freedom usuallyimplies softening of the EoS, and consequently, eventu-ally obtained maximal masses are far below the 2 M.This leads, for instance, to the famous hyperon puzzle[48]. One of the most prominent solutions to this prob-lem would be a strong phase transition to deconfinedquark matter [49, 50]. Sufficient stiffness of the EoS athigh densities is required in order to obtain massive solu-tions of the Tolman Oppenheimer Volkoff (TOV) equa-tions [51, 52]. In phenomenological models, it is typicallyprovided by repulsive interactions mediated by the ex-change of vector mesons. The inclusion of repulsive vec-tor interactions in quark matter allows for hybrid starsthat fulfill the 2 M constraint, as demonstrated in [53],[54], [55], and [56]. They are also known to be impor-tant for modeling the QCD phase diagram [57]. At mod-erate densities, however, the quark matter EoS shouldbe soft enough such that the transition to the decon-fined phase eventually takes place. Therefore, a naturalquestion to ask is whether or not massive NSs, such asthe recently discovered PSR J0740+6620 with mass of2.14+0.20

−0.18 M [36], could contain deconfined quark mat-ter in their core. This problem has attracted a lot ofattention over the past two decades (see, e.g., [53, 58–65]), but no decisive observational evidence has yet beenprovided.

The mechanism of quark confinement and its relationto the chiral symmetry breaking are of major importancein probing the hadron-quark phase transition, althoughit is nontrivial to embed their interplay into a single ef-

fective theory. For this reason, despite its serious short-comings, the conventional approach is to use separateeffective models for the nuclear and quark matter phases(two-phase approaches) with a priori assumed first-orderphase transition, typically associated with simultaneouschiral and deconfinement transitions [66]. Major at-tempts to go beyond the two-phase approach are, for ex-ample, the Polyakov loop-extended Nambu–Jona-Lasinio(PNJL) models [67–71], or the Polyakov loop-extendedquark-meson (PQM) [72–76]. While Polyakov loop-basedmodels provide dynamical means to mimic confinementof quarks, their caveat is the lack of hadrons as degrees offreedom at low temperature and/or density. Moreover,while at finite temperature and low density the Polyakovloop expectation value serves as an approximated orderparameter for deconfinement, it is highly questionablethat it remains so at high density, which is relevant forastrophysical applications.

In this work, we provide the effective dynamical chiralmodel to quantify the high-density EoS in the presence ofcolor deconfinement and the chiral symmetry restoration.Instead of using the Polyakov-loop-based model, we em-ploy the hybrid quark-meson-nucleon (QMN) model [16–19] to explore the implications of dynamical sequentialphase transitions at high baryon density on the phe-nomenology of neutron stars. The model has the charac-teristic feature that, upon increasing baryon density, thechiral symmetry is restored within the hadronic phase bylifting the mass splitting between chiral partner states,before the quark deconfinement takes place. Quark de-grees of freedom are included on top of hadrons, but theirunphysical onset is prevented at low densities. This isachieved by an auxiliary scalar field which couples toboth nucleons and quarks. This field serves as a mo-mentum cutoff in the Fermi-Dirac distribution functions,thus it suppresses the unphysical thermal fluctuations offermions, with the strength linked to the density.

The previous implementation of the hybrid QMNmodel to obtain the nuclear matter EoS leads to the con-clusion, that the hadronic branch of the mass-radius rela-tion is at tension with the two-solar-mass constraint [18,19]. In order to improve the description of quark matterproperties, we extended the previous hybrid QMN modelby including an isoscalar-vector and isovector-vector cou-pling to quarks. We systematically study the role ofthe repulsive quark-vector coupling on the deconfinementphase transition and phenomenology of compact stars.

The hybrid QMN model naturally embeds the inter-play between the quark confinement and the chiral sym-metry breaking into a single field-theoretical framework,which makes the phenomena inherently connected, incontrast to the two-phase approach. These key phe-nomenological ingredients of the model may have impor-tance not only in the context of modeling of the interiorof NSs, but also in dynamical simulations of their merg-ers, as well as supernova collapse and explosions. Previ-ous studies have pointed out the importance of the QCDphase transitions, where the additional latent heat can

3

trigger the local production of neutrinos [31, 32, 56, 77–81]. This also motivates further study and possible ap-plications of the proposed model in the context of HICat finite temperature, where a novel signature of chiralsymmetry restoration within the dense hadronic sectorin dilepton production was recently proposed [82].

This paper is organized as follows. In Sect. II, we intro-duce the hybrid quark-meson-nucleon model. In Sect. III,we discuss the obtained numerical results on the equationof state under neutron-star conditions. In Sect. IV, wediscuss the obtained neutron-star relations and confrontthe results with recent observations. Finally, Sect. V isfeatures our summary and conclusions.

II. HYBRID QMN MODEL

In this section, we briefly introduce the hybrid QMNmodel, which is capable of describing the chiral symme-try restoration and deconfinement phase transitions [16–19]. The hybrid QMN model is composed of the bary-onic parity doublet [12–14] and mesons as in the Waleckamodel [83], as well as quark degrees of freedom as in thestandard linear sigma model [84]. The spontaneous chi-ral symmetry breaking yields the mass splitting betweenthe two baryonic parity partners, while it generates theentire mass of a constituent quark. In this work, we con-sider a system with Nf = 2; hence, relevant for this studyare the positive-parity nucleons, which are proton (p+)and neutron (n+), and their negative-parity partners, de-noted as p− and n−, as well as the up (u) and down (d)quarks. The fermionic degrees of freedom are coupled tothe chiral fields (σ,π), the isosinglet vector-isoscalar field(ωµ), and the vector-isovector field (ρµ). The importantconcept of statistical confinement is realized in the hy-brid QMN model by introducing a medium-dependentmodification of the particle distribution functions.

The Lagrangian of the model reads

L = LN + LM + Lq, (1)

with LN , LM , Lq denoting the nucleon, meson, andquark parts, respectively. The nucleon part (LN ) of theLagrangian reads

LN = iψ1/∂ψ1 + iψ2/∂ψ2 +m0

(ψ1γ5ψ2 − ψ2γ5ψ1

)+g1ψ1 (σ + iγ5τ · π)ψ1 + g2ψ2 (σ − iγ5τ · π)ψ2

−gNω∑k=1,2

ψk/ωψk −1

2gNρ

∑k=1,2

ψkτ · /ρψk,

(2)

where ψk is the set of baryonic chiral fields. Parametersg1, g2, and gNω , gNρ are the baryon-to-meson couplingconstants, and m0 is a mass parameter.

The mesonic part, LM , of the Lagrangian is introducedas

LM =1

2(∂µσ)

2+

1

2(∂µπ)

2 − 1

4(ωµν)

2 − 1

4

(ρµν

)2− Vσ − Vω − Vρ,

(3)

where ωµν = ∂µων − ∂νωµ is the field-strength tensor ofthe vector-isoscalar field, ρµν = ∂µρν−∂νρµ−gNρ ρµ×ρνis the field-strength tensor of the vector-isovector field,and the potentials read

Vσ = −λ22

Σ +λ44

Σ2 − λ66

Σ3 − εσ, (4a)

Vω = −m2ω

2ωµω

µ, (4b)

Vρ = −m2ρ

2ρµρ

µ, (4c)

where Σ = σ2 + π2, λ2 = λ4f2π − λ6f4π − m2

π, and ε =m2πfπ. mπ, mω, and mρ are the π, ω, and ρ meson

masses, respectively, and fπ is the pion decay constant.The parameters λ4 and λ6 are fixed by the propertiesof the nuclear ground state. We note that the six-pointscalar interaction term in Eq. (4a) is essential in order toreproduce the experimental value of the compressibilityK = 240± 20 MeV [85, 86]. Numerical values of modelparameters are summarized in Tables I, II, and III.

In the physical basis, the effective masses of the chiralpartners, mp± = mn± ≡ m±, are given by

m± =1

2

[√(g1 + g2)

2σ2 + 4m2

0 ∓ (g1 − g2)σ

]. (5)

The positive-parity nucleons are identified as the posi-tively charged and neutral N(938) states: proton (p+)and neutron (n+). Their negative-parity counterparts,denoted as p− and n−, are identified as N(1535) [87].From Eq. (5), it is clear that the chiral symmetry break-ing generates only the splitting between the two masses.

When the chiral symmetry is restored, the masses ofthe baryonic parity partners become degenerate with acommon finite mass m± (σ = 0) = m0, which reflects theparity doubling structure of the low-lying baryons. Fol-lowing the previous studies of the parity-doublet-basedmodels [15–30], as well as recent LQCD results [10, 11],we choose a rather large value: m0 = 700 MeV. Thecouplings g1, g2 in Eq. (5) can be determined by fixingthe fermion masses in the vacuum. Their values used inthis work are summarized in Table I. Having fixed thechirally invariant mass m0, we obtain the values of theeffective masses at saturation density, which are denotedas m∗

±. We find that the effective mass of the positive-parity states, m∗

+, lies within the experimentally deter-mined range [46, 88]. We note, however, that their exactvalues depend on the choice of the parameter m0 [15].Thus, in principle, the experimental value of m∗

+ couldbe used as an input parameter to constrain the modelparameters. In Table II, we show the obtained values ofthe effective masses at saturation density for both paritypartners.

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m+ [MeV] m− [MeV] mπ [MeV] fπ [MeV] mω [MeV] mρ [MeV]

939 1500 140 93 783 775

TABLE I. Physical vacuum inputs used in this work.

ρ0 [fm−3] E/A−m+ [MeV] K [MeV] Esym [MeV] L [MeV] m∗+ [mvac

+ ] m∗− [mvac

− ]

0.16 -16 240 31 82 0.82 0.76

TABLE II. Properties of the nuclear ground state at µB = 923 MeV and the symmetry energy used in this work. Alsotabulated is the slope of the symmetry energy and the effective masses of the positive- and negative-parity baryonic states atthe saturation density given in the units of their vacuum masses.

λ4 λ6f2π gNω gNρ g1 g2 gq κb [MeV] λb b0 [MeV]

33.74 13.20 7.26 7.92 13.75 7.72 3.36 155 0.074 569.79

TABLE III. Numerical values of the model parameters. The values of λ4, λ6, and gNω are fixed by the nuclear ground stateproperties, gNρ by the symmetry energy, and gq is fixed by the vacuum quark mass (see the text). The remaining parameters,κb, λb, and b0 are fixed following [17].

The quark part(Lq) of the Lagrangian in Eq. (1) reads

Lq =∑k=u,d

iψk/∂ψk + gqψk (σ + iγ5τ · π)ψk

− gqω∑k=u,d

ψk/ωψk −1

2gqρ∑k=u,d

ψkτ · /ρψk,(6)

where ψk is a set of fields for up (u) and down (d)quarks. Parameters gqσ, gqω and gqρ are the quark-to-meson coupling constants. The quark effective mass,mu = md ≡ mq, is linked to the sigma field as

mq = gqσ. (7)

We note that in contrast to the baryonic parity part-ners (cf. Eq. (5)), quarks become massless as the chiralsymmetry gets restored. The value of the coupling gq inEq. (7) can be determined by assuming the quark massto be mq = 1/3 m+ in the vacuum.

The strength of gNω is fixed by the nuclear saturationproperties, while the value of gNρ can be fixed by fittingthe value of symmetry energy [46]. The properties ofthe nuclear ground state and the symmetry energy areshown in Table II. On the other hand, the nature of therepulsive interaction among quarks and their couplingto the ω and ρ mean fields are still far from consensus.To account for the uncertainty in the theoretical predic-tions, we treat the couplings gqω and gqρ as free parame-ters. We parametrize the couplings of quarks to the ωand ρ mesons with a single dimensionless parameter χ asfollows:

gqω = χgNω , (8a)

gqρ = χgNρ . (8b)

We stress that the original hybrid QMN model wasdesigned to saturate the Stefan-Boltzmann limit at hightemperature and density [16, 17]. However, the intro-duction of an additional quark-vector coupling spoils thislimit. On the other hand, in this work we considered vec-tor fields only up to quadratic order (cf. Eq. (4)), thus,the resulting terms in the effective mean-field potentialare also up to quadratic order in densities and do notlead to the acausal behavior of the sound velocity, mean-ing c2s ≡ ∂P/∂ε < 1 is fulfilled [89]. We note that it isgenerally possible to sustain the 2 M constraint and ful-fill the conformal bound,c2s ≤ 1/3. This can be obtained,for example, in a class of constant-speed-of-sound equa-tions of state [90].

The hybrid QMN model realizes the concept of statis-tical confinement through a medium-dependent modifi-cation of the Fermi-Dirac distribution functions, wherean auxiliary scalar field, b (bag field), is introduced. Thedistribution functions for the quarks and antiquarks arereplaced with

nq = θ(p2 − b2

)fq, (9a)

nq = θ(p2 − b2

)fq, (9b)

where b is the expectation value of the b-field. The dis-tribution functions for the nucleons and antinuclones arereplaced with

n± = θ(α2b2 − p2

)f±, (10a)

n± = θ(α2b2 − p2

)f±, (10b)

where α is a dimensionless model parameter, and f, fare the standard Fermi-Dirac distribution functions forthe particle and antiparticle, respectively. As we demon-strate in Sect. III, the parameter α plays a crucial role intuning the order of the chiral phase transition [16, 17].

5

From the definition of n± and nq, it is evident thatin order to mimic the statistical confinement, the b fieldshould have a nontrivial vacuum expectation value, so asto suppress quark degrees of freedom at low densities inthe confined phase and to allow for their population athigh densities in the deconfined phase. This is achievedby allowing b to be generated from the potential Vb. Fol-lowing [16], we take the potential of the minimal form

Vb = −1

2κ2bb

2 +1

4λbb

4. (11)

The potential Vb for a positive κ2b develops a nontriv-

ial vacuum expectation value at b0 =√κ2b/λb. From

Eqs. (9) and (10), one finds that the nucleons favor largeb, whereas the quarks favor small b. The potential (Eq.11) is chosen such that, at a certain T and µB , a tran-sition sets in, causing the bag-field expectation value toabruptly drop. As a consequence, at low T and µB , thequark degrees of freedom are suppressed, while the nucle-ons get suppressed at high T and µB . This characteristicbehavior is associated with the deconfinement transition,which is a crucial feature of the model [16].

In Eqs. (9) and (10), the functions fx and fx are thestandard Fermi-Dirac distributions,

fx =1

1 + eβ(Ex−µx), (12a)

fx =1

1 + eβ(Ex+µx), (12b)

with β being the inverse temperature and the dispersionrelation Ex =

√p2 +m2

x. The effective chemical poten-

tials for p± and n± are defined as#1

µp± = µB − gNω ω −1

2gNρ ρ+ µQ, (13a)

µn± = µB − gNω ω +1

2gNρ ρ. (13b)

The effective chemical potentials for up and down quarksare given by

µu =1

3µB − gqωω −

1

2gqρρ+

2

3µQ, (14a)

µd =1

3µB − gqωω +

1

2gqρρ−

1

3µQ. (14b)

In Eqs. (13) and (14), µB , µQ are the baryon and chargechemical potentials, respectively.

The concept of momentum cutoff is naturally sup-ported by the asymptotic freedom in QCD, which sug-gests that active degrees of freedom are different de-pending on their momenta, meaning hadrons (quarks)

#1 In the mean-field approximation, the nonvanishing expectationvalue of the ω field is the time-like component; hence we simplydenote it by ω0 ≡ ω. Similarly, we denote the nonvanishingcomponent of the ρ field, time-like and neutral, by ρ03 ≡ ρ.

with low (high) momenta. Such notion has been widelyused in effective theories and the Dyson-Schwinger ap-proach [91, 92]. In [16], this idea was pushed further byintroducing a momentum cut-off as a medium-dependentquantity. Such an intrinsic modification of the cut-off isdetermined self-consistently when the cut-off is regardedas an expectation value of a scalar field, which is b in ourmodel. As demonstrated in [16], the b field plays a rolesimilar to that of the Polyakov loop at finite tempera-ture and vanishing chemical potential, and it is respon-sible for the onset of quark degrees of freedom aroundthe crossover temperature. Although the b field is so farnot anchored to any QCD symmetry, its introduction isa minimal and practical way to realize the physical situa-tion where quarks are not activated deep in the hadronicphase. Therefore, b is likely related to gluons, especiallythe chromoelectric component. So far it is not knownhow one is connected to another, since there is no rigor-ous order parameter of confinement in QCD with lightfermions. Thus, the b field cannot be regarded as a trueorder parameter. In recent studies, it has been shownthat such nontrivial order parameter of the center flavorsymmetry exists in a QCD-like theory compactified on acircle [93, 94].

The parameter α manifests itself only in combinationwith the expectation value of the b field. Therefore, αrelates the scales for the momentum distributions of nu-cleons (αb) and quarks (b). The lower bound for the αparameter is set so that the ultraviolet cut-off αb0 fornucleon distribution functions and the infrared cut-off b0for quark distribution functions do not spoil the prop-erties of the nuclear ground state. This sets the lowerbound to be roughly αb0 ≥ 300 [16, 17]. On the otherhand, the upper bound on the parameter can be de-rived by requiring that the system correctly saturatesthe Stefan-Boltzmann limit, which yields the maximalvalue for αb0 ≤ 450 MeV [16, 17]. Following our pre-vious works, we chose four representative values withinthat interval: αb0 = 350, 370, 400, and 450 MeV, so asto systematically study the influence of the quark-vectorcouplings on the equation of state and the resulting mass-radius relations.

Intuitively, the meaning behind the momentum cut-offb and the α parameter can be understood within a simplebag model picture, where the nucleon is a bound state ofthree noninteracting massless valence quarks, which areconfined in a bag of radius r. Due to the uncertaintyprinciple, the quarks cannot have momenta lower thanq0 ≥ 1/r. With the mass of the ground state nucleonbeing m+ = 3q0 = 939 MeV, this yields the value ofthe radius r = 0.63 fm. When the nuclear matter isdense enough, the nucleon wave functions start overlap-ping and the system exhibits the Pauli blocking effect. Inorder to estimate the density at which the Pauli blockingmanifests itself, one can use the van der Waals excludedvolume of hard-sphere nucleons [96]:

vex =16

3πr3, (15)

6

10

100

1000

2 4 8 16

10

100

2 4 6

P[M

eV/fm

3]

ρB [ρ0]

Danielewicz et alχ = 0.2χ = 0.1χ = 0

10

100

1000

2 4 8 16

m0 = 700 MeVαb0 = 370 MeV

αb0 = 350 MeVαb0 = 370 MeVαb0 = 400 MeVαb0 = 450 MeV

10

100

2 4 610

100

1000

2 4 8 16

1

10

100

1000

100 1000

P[M

eV/fm

3]

ρB [ρ0]

χ = 0.2χ = 0.1χ = 0

10

100

1000

2 4 8 16

m0 = 700 MeVαb0 = 370 MeV

P[M

eV/fm

3]

ε [MeV/fm3]

Hebeler et alαb0 = 350 MeVαb0 = 370 MeVαb0 = 400 MeVαb0 = 450 MeV1

10

100

1000

100 1000

FIG. 1. Thermodynamic pressure, P , for isospin-symmetric matter (left panel) and under the NS conditions of β-equilibriumand charge neutrality (right panel), as functions of the net-baryon number density, ρB , in units of the saturation density,ρ0 = 0.16 fm−3 for m0 = 700 MeV, αb0 = 370 MeV and different values of the parameter χ (see text for details). In the leftpanel, the flow constraint [95] is shown as the gray shaded region. In both panels, the inset plots show pressure for χ = 0 anddifferent values of the parameter α, in the vicinity of the constraints obtained by [95] (left panel) and [42] (right panel). Wenote that the inset figure in the right panel shows pressure as a function of energy density. In both panels, the first-order phasetransitions are seen as plateaux of constant pressure.

which corresponds to the closest packing density of ρc =1/vex = 0.238 fm−3 = 1.49ρ0, where ρ0 = 0.16 fm−3 isthe saturation density. The density ρc naturally corre-sponds to the onset of the stiffening effect in the hybridQMN model. At higher density, the wave functions ofnucleons overlap and the quark momenta have to be re-arranged, in order not to violate the Pauli exclusion prin-ciple. A simple argument how the Pauli principle may actin a dense gas of nucleons as three-quark bound states isthat the higher quark momenta have to be populated.At zero temperature, the shell in the quark-momentumspace, ∆ρq, which has to be filled, ranges from q0 toq0 + δq0 and corresponds to the quark density

∆ρq = 3∆ρB = 3γq

q0+δq0∫q0

dqq2

2π2=

γq2π2

q20δq0 +O(δq20),

(16)where γq is the spin-color-flavor degeneracy factor ofquarks. From the above, the shift of the quark momen-tum, δq0, can be related to the corresponding Pauli shiftin the nucleon energy, ∆Pauli = 3δq0 = 18π2/(γqq

20)∆ρB .

At densities exceeding the critical density, ρc, where theeffect sets in, the Pauli shift corresponds to a density-dependent lowering of the chemical potential. The crit-ical density can be translated to the critical Fermi mo-mentum of nucleons,

pf =3

√3

2π2ρc =

3

√9

32π q0, (17)

and compared with the corresponding UV cut-off αb0 inthe hybrid QMN model. The value q0 is identified withthe IR cut-off in the quark momentum distribution due

to confinement and the uncertainty relation. In the hy-brid QMN model, it is given by b0. Thus, one gets theUV cut-off in the nucleon distribution to be pf = αb0.

From this, one gets α = 3√

9π/32 ≈ 0.959. We note thatthe maximal value of the α parameter obtained in thehybrid QMN model is αmax = 2−1/3 ≈ 0.794 [16, 17].This simple analysis serves as a ballpark estimate for theparameter α and shows that it may be quantified by con-sidering the Pauli exclusion principle on the quark level.The present estimate within an intuitive bag model pic-ture and the van der Waals excluded volume leads tosimilar scaling as the one obtained in the hybrid QMNmodel, meaning smaller values of b0 and q0 lead to anearlier onset of the stiffening effect. A more realistic cal-culation of the quark Pauli blocking in nuclear mattercan be found in [97].

The thermodynamic potential in the mean-field ap-proximation reads [18]

Ω =∑

x=p±,n±,u,d

Ωx + Vσ + Vω + Vρ + Vb, (18)

where the summation goes over the fermionic degrees offreedom. The kinetic part, Ωx, of the thermodynamicpotential Eq. (18), reads

Ωx = γx

∫d3p

(2π)3T [ln (1− nx) + ln (1− nx)] , (19)

where the functions nx and nx are the modified Fermi-Dirac distributions defined in Eqs. (9) and (10). The spindegeneracy factor, γx for nucleons is γ± = 2 for bothpositive- and negative-parity states, while the spin-colordegeneracy factor for quarks is γq = 2× 3 = 6.

7

The physical inputs and the model parameters used inthis work are summarized in Tables I and II. In-mediumprofiles of the mean fields are obtained by extremizingthe thermodynamic potential in Eq. (18), leading to thefollowing gap equations:

∂Ω

∂σ=∂Vσ∂σ

+∑

x=p±,n±,u,d

sx∂mx

∂σ, (20a)

∂Ω

∂ω=∂Vω∂ω

+∑

x=p±,n±,u,d

gxωρx, (20b)

∂Ω

∂ρ=∂Vρ∂ρ

+∑

x=p±,u

gxρ2ρx −

∑x=n±,d

gxρ2ρx, (20c)

∂Ω

∂b=∂Vb∂b

+ α∑

x=p±,n±

ωx −∑x=u,d

ωx, (20d)

where the scalar and baryon densities are

sx = γx

∫d3p

(2π)3mx

Ex(nx + nx) (21)

and

ρx = γx

∫d3p

(2π)3(nx − nx) , (22)

respectively. The terms ωx in the gap Eq. (20d) are givenas

ω± = γ±(αb)2

2π2T[ln (1 − f±) + ln

(1 − f±

)]p2=(αb)2

(23)

and

ωq = γqb2

2π2T[ln (1 − fq) + ln

(1 − fq

)]p2=b2

(24)

for the nucleons and quarks, respectively. We note thatthe terms in Eqs. (23) and (24) come into the gapEq. (20d) with opposite signs. This reflects the fact thatnucleons and quarks favor different values of the bag field.

In the grand canonical ensemble, the thermodynamicpressure is obtained from the thermodynamic potentialas P = −Ω + Ω0, where Ω0 is the value of the thermody-namic potential in the vacuum. The net-baryon numberdensity for a species x is defined as

ρxB = −∂Ωx∂µB

, (25)

where Ωx is the kinetic term in Eq. (19). The total net-baryon number density reads

ρB = ρn+

B + ρn−B + ρ

p+B + ρ

p−B + ρuB + ρdB . (26)

The particle-density fractions are defined as

Yx =ρxBρB

. (27)

The symmetry energy and its slope at saturation densityare given as

Esym =1

2

∂2 (ε/ρB)

∂δ2

∣∣∣∣∣δ=0

(28)

and

L = 3ρ0∂Esym

∂ρB

∣∣∣∣∣ρB=ρ0

, (29)

respectively. In Eq. (28), ε is the en-ergy density and the asymmetry parameter

δ = (ρn+

B + ρn−

B − ρp+

B − ρp−

B + ρdB − ρuB)/ρB#2. The

value of the slope of the symmetry energy at saturationdensity obtained in the model is L = 82 MeV. Thevalue is rather large when compared to the recentchiral effective field theory estimate [98]. We note thatsimilar values are found in other parity-doublet modelsat the mean-field-level thermodynamics [86], and it isconsistent with the commonly considered range for theparameter [47, 99].

In the following section, the above hybrid QMN modelequation of state of strongly interacting matter will beapplied to identify properties of compact stellar objectssuch as NSs.

III. EQUATION OF STATE

We started by generalizing the hybrid QMN EoSderived for the cold nuclear matter to account forthe isospin effects, considering isospin-symmetric andhighly asymmetric environments such as the interiors ofNSs. The composition of neutron-star matter requiresβ equilibrium with electrons (e) and muons (µ), includedas free relativistic particles, as well as the charge neutral-ity condition. To this end, we solve the set of equations,

µe = µµ = µn+ + µn− − µp+ − µp− + µd − µu (30a)

ρp+ + ρp− +2

3ρu −

1

3ρd − ρe − ρµ = 0, (30b)

at T = 0, where µx’s and ρx’s are the effective chemicalpotentials of given species (see Eqs. (13) and (14)) andtheir densities (see Eq. (22)), respectively.

In Fig. 1, we show the calculated zero-temperatureequations of state in the mean-field approximation forselected parametrization, m0 = 700 MeV and αb0 =370 MeV, for different strengths of the quark vector-interaction χ (see Eq. (8)), namely χ = 0 (red, solid line),

#2 We note that this generalized definition reduces to the known def-

inition of the asymmetry parameter, δ = (ρn+

B − ρp+

B )/ρB , wherethe negative parity states and quarks are not populated and theirdensities vanish, which is the case at the saturation density.

8

αb0 [MeV]

χ 350 370 400 450

01.82 − 2.60 2.14 − 2.76 2.61 − 2.92 3.56

4.98 − 6.11 5.84 − 6.21 5.10 4.82 − 6.04

9.21 − 13.58 11.03 − 15.42 16.40 − 19.25 10.84

0.11.82 − 2.60 2.14 − 2.76 2.61 − 2.92 3.56

6.56 − 7.20 6.82 − 7.75 7.13 − 8.32 6.61

8.48 − 12.32 9.86 − 13.54 11.96 − 15.89 15.67 − 19.63

0.21.82 − 2.60 2.14 − 2.76 2.61 − 2.92 3.56

12.07 − 15.67 11.19 − 11.51 11.97 − 12.59 11.89 − 12.82

12.07 − 15.67 14.38 − 18.00 17.89 − 21.72 23.28 − 27.56

TABLE IV. Baryon density ranges of the coexistence phases associated with the chiral restoration (top), onset of down (middle)and up (bottom) quark under the neutron-star conditions, in terms of saturation density units, ρ0, for different values of αb0and χ parameters. In the cases where transitions proceed as smooth crossovers, a single value is given.

χ = 0.1 (purple, dashed line), and χ = 0.2 (blue, dottedline), as functions of the baryon density in the units ofsaturation density. In the left panel of the figure, we showthe isospin-symmetric matter EoSs. In each case, the be-havior at low density is similar. Shown EoSs feature acommon first-order chiral phase transition, determinedas a jump in the σ-field expectation value, which causesthe parity-doublet nucleons to become almost degener-ate with the mass m± = m0. The chiral phase transitionis triggered at roughly 3.25 ρ0, with the mixed phasepersisting up to 3.68 ρ0. The mechanism of statisticalconfinement has a prominent twofold impact on the classof EoSs obtained in the model. First, the strength ofthe chiral phase transition depends on the choice of α.Namely, higher values yield weaker first-order transition,which turns into a second-order and eventually becomesa smooth crossover, defined as a peak in ∂σ/∂µB . Sec-ond, lower values of α result in a stiffening of the EoS,due to which, the chiral phase transition is triggered atlower densities. [17]. We illustrate the stiffening of theEoS based on a simplified model in Appendix A.

We note that the stiffening effect of nuclear matter justabove the saturation density is a feature that the presentapproach has in common with, for example, the relativis-tic density-functional models of nuclear matter with ex-cluded nucleon volume (such as the one by [100]). Theseare used, for example, in the effective multi-polytrope[44] and CSS [101] class of hybrid EoS. However, too ex-treme stiffening could represent a certain tension withthe recent analysis of GW170817 [37]. In principle, thistension could be resolved, for example, by a strong phasetransition occurring in the compact star mass range rel-evant for GW170817 (see reference [102]).

In the left panel of Fig. 1, we also show the protonflow constraint (gray region) obtained in heavy ion col-lisions [95], where the matter is assumed to be isospin-symmetric. The agreement with the EoSs obtained inthe hybrid QMN model is fairly good. Shown EoSsgo through the upper band of the constraint. Inter-

estingly, the chiral phase transition happens within thebaryon-density range of the flow constraint. The in-set plot in the left panel of Fig. 1 shows the EoSs ob-tained for different values of the α parameter in thevicinity of the flow constraint. Equations of state forαb0 = 350, 370, and 400 MeV feature the first-order chi-ral phase transition within this constraint. The softestEoS for αb0 = 450 MeV features a smooth crossover tran-sition slightly above the densities governed by the flowconstraint, at roughly 5.4 ρ0. The stiffening of the EoSsis evidently seen before the chiral phase transition takesplace. This highlights the importance of the chiral phasetransition for the density range accessible in heavy ioncollisions. We note that the chiral symmetry restorationin the dense hadronic sector can possibly be identifiedin the production rate of dileptons, as proposed recentlyin [82].

As the density increases, the equation of state with noquark-vector coupling, that is, χ = 0, features anotherjump in baryon density, associated with the deconfine-ment of quarks, with a mixed phase from 9.65−16.04 ρ0.When the inclusion of the quark-vector coupling is con-sidered, meaning the χ = 0.1 and χ = 0.2 cases in Fig. 1,the onset of both quarks is systematically shifted towardhigher densities, when compared to the case with vanish-ing coupling, with mixed phases between 11.19−17.28 ρ0and 19.18 − 24.83 ρ0, respectively. Consequently, thehadronic part of the EoS is extended beyond the decon-finement in the case of vanishing χ. We note that allthree equations of state remain the same up to the pointwhere the quarks start to appear. We stress that the chi-ral and hadron-to-quark phase transitions are sequential.

In the right panel of Fig. 1, we show the correspond-ing EoSs under the neutron-star conditions for m0 =700 MeV and αb0 = 370 MeV. Similarly to the isospin-symmetric case, the behavior at low densities is quitecomparable. Shown EoSs feature a common first-orderchiral phase transition. For comparison, the EoSs ob-tained for the remaining values of the parameter α are

9

0.1

0.3

0.5

0.7

1 2 4 8 16

0.1

0.3

0.5

0.7

0.1

0.3

0.5

0.7

ρB [ρ0]

0.1

0.3

0.5

0.7

1 2 4 8 16

n+

p+

e

µ

n−

p−

d

u

Mmax = 2.08 M

χ = 0.2

Yx

0.1

0.3

0.5

0.7 n+

p+

e

µ

n−

p−

d

u

Mmax = 2.08 M

χ = 0.1

0.1

0.3

0.5

0.7 n+

p+

e

µ

n−

p−

d

u

Mmax = 2.06 M

χ = 0

FIG. 2. Composition of compact star matter as a function ofthe central baryon density in the units of saturation densityfor the present model for m0 = 700 MeV and αb0 = 370 MeV.Three different cases are considered for χ = 0 (top panel),χ = 0.1 (middle panel), χ = 0.2 (bottom panel). In eachpanel, the yellow shaded areas indicate the density regionsof phase coexistence. The gray, solid vertical lines indicatethe central densities of the associated maximum-mass neu-tron stars.

also shown in the vicinity of the chiral phase transitions.We note that the chiral phase transitions occur at sys-tematically lower densities in comparison to the isospin-symmetric matter. However, as the density increases,the EoSs feature another two sequential jumps in baryondensity. The first is associated with the onset of thedown quark, and the second is associated with the on-set of the up quark, after which the EoS is composedsolely of quarks. The sequential appearance of quarks inthe isospin-asymmetric matter stays in contrast to thecase of symmetric case, where quarks deconfine simul-taneously, owing to the isospin symmetry. This can betraced back to their different electric charge, as well asto the coupling to the ρ meson, which introduces dif-ferent signs for both quarks in their effective chemicalpotentials (see Eq. (14)). Thus, the obtained EoSs un-der the neutron-star conditions feature sequential (i.e.,flavor-dependent) deconfinement phase transition. We

stress that the sequential deconfinement of quarks un-der the neutron-star conditions cannot be obtained in aclass of pure quark models or even in a class of hadron-quark models, where the confined and deconfined phasesare treated independently. This is due to the negativecharge of the down quark, which cannot be neutralizedwith negatively-charged leptons. The effect of the finitequark-vector coupling is similar to the isospin-symmetriccase. Namely, the onset of both quarks is systemati-cally shifted toward higher densities when compared tothe case with vanishing coupling. Consequent extensionof the hadronic branch of the EoS is exhibited. In thiscase, the EoSs remain the same up to the point wherethe down quark appears. This behavior is also resembledby the density fractions shown in Fig. 2. The composi-tion of the compact-star matter for m0 = 700 MeV andαb0 = 370 MeV and χ = 0 (top panel), χ = 0.1 (middlepanel), χ = 0.2 (bottom panel), in terms of the particle-density fractions, Yx, defined in Eq. (27), as a functionof the baryon density in the units of saturation density isalso illustrated in Fig. 2. Similarly to the EoS, the com-positions at low density are the same, up to the point ofthe appearance of the down quark in the case with χ = 0(green, dash-dotted line in the top panel). With a finitevalue of the parameter χ, the hadronic part of the EoS isextended, and consequently the deconfinement of quarksis shifted to higher densities. Because the down quark al-ways appears before the up quark, due to lower Fermi mo-menta, in order to fulfill the charge neutrality constraint,its negative electric charge can only be neutralized by thepositively-charged hadrons, meaning protons, p+, and itschiral partners, p−. Such a mixed phase, comprised ofhadrons and the down quark, persists until the hadronsbecome suppressed and the up quark is populated, com-pensating for the positively-charged hadrons in order tomeet the charge neutrality condition. We note that inthe case of isospin-asymmetric matter, the phase transi-tions are triggered at lower densities. The baryon-densityjumps associated with consecutive sequential transitionsfor the sets of model parameters used in this work areshown in Tab. IV.

In Fig. 4, we show the speed of sound squared,c2s = dP/dε, in units of the speed of light squared, asa function of the net-baryon number density. The co-existence phases due to first-order phase transitions areseen as plateaux of vanishing speed of sound. The no-table swift increases of the speed of sound in each curveare results of the stiffening mechanism that arises due tothe statistical confinement implemented in the model (cf.Eq. (10)).

We note that there is a theoretical uncertainty in theprediction for the ratio of the vector- and scalar-quarkcouplings in the class of Nambu–Jona-Lasinio (NJL) andthe Polyakov loop-extended NJL models [54, 63, 64, 71,103, 104]. Interestingly, we find that the results are es-sentially independent of the exact value of the scalar cou-pling, gq in the hybrid QMN model. This is because thevalue of the coupling is fixed to the the properties in the

10

1

10

100

1000

100 1000

P[M

eV/fm

3]

ε [MeV/fm3]

Hebeler et alαb0 = 450 MeVαb0 = 400 MeVαb0 = 370 MeVαb0 = 350 MeV

1

10

100

1000

100 1000

m0 = 700 MeV

FIG. 3. Thermodynamic pressure, P , under the NS condi-tions of β-equilibrium and charge neutrality, as a function ofthe energy density, ε, for m0 = 700 MeV and χ = 0, and fourrepresentative values of the parameter α (see text for details).The phase transitions are seen as plateaux of constant pres-sure. The gray shaded region marks the constraint obtainedby [42].

vacuum, where quarks are not expected to be present aseffective degrees of freedom. In the hybrid QMN model,this is controlled by the finite expectation value of theb-field, which suppresses the quark contribution to thegap equations at low density (see Sect. II). On the otherhand, the quarks are populated at high densities. There,however, the expectation value of the scalar mean field,σ, is of the order of a few MeV, and slowly vanishes asthe density increases. Therefore, the quark contributionto the gap equation in the σ direction is negligible. Thus,contrary to other effective models, the laxity in the valueof the coupling, gq, is a direct consequence of the confine-ment mechanism implemented in the model, and seemsnatural from the phenomenological point of view. Wenote that at high temperatures it might not persist, ow-ing to additional thermal fluctuations.

We stress that all EoSs discussed in this work fall intothe region derived from the maximum mass constraintMmax ≥ 2.01(4) M [34] obtained by Hebeler et al inRef. [42] using a multi-polytrope ansatz for the EoS abovethe saturation density. This is shown in Fig. 3 for thecase of vanishing parameter χ. Interestingly, the chiralphase transition and the deconfinement of down quark liewithin the region set by the constraint. This is in contrastto the symmetric-matter flow constraint, where only thechiral transition is featured within the constraint. Theup quarks are populated only at higher values of the en-ergy density. We note that for high enough value of theparameter χ, the deconfinement transition would shift tohigher density and eventually fall out of the constraint.

In general, the class of EoSs obtained in the hybridQMN model for the case of isospin-asymmetric matterconsist of four consecutive density regions, from low to

0

0.2

0.4

0.6

0.8

2 4 8 16

c2 s

ρB [ρ0]

χ = 0.2χ = 0.1χ = 0

0

0.2

0.4

0.6

0.8

2 4 8 16

αb0 = 370 MeVm0 = 700 MeV

FIG. 4. Speed of sound squared as a function of the net-baryon number density, ρB , in units of the saturation density,ρ0 = 0.16fm−3 for m0 = 700 MeV, αb0 = 370 MeV anddifferent values of the parameter χ (see text for details). Thefirst order phase transitions are seen as plateaux of vanishingspeed of sound.

αb0 [MeV]

χ 350 370 400 450

0 2.11, 11.89 2.06, 11.72 2.01, 11.64 1.98, 11.74

0.1 2.13, 11.43 2.08, 11.22 2.03, 11.11 2.01, 11.27

0.2 2.13, 11.43 2.08, 11.22 2.03, 11.11 2.01, 11.27

TABLE V. Maximal neutron-star masses in units of M andcorresponding radius in km (separated by comma) for differ-ent values of αb0 and χ parameters for m0 = 700 MeV.

high density: confined and chirally broken phase, con-fined and chirally restored phase, partially deconfined,and fully deconfined phase. The intermediate phases arefor the novel states in which confined and deconfined mat-ter may coexist. Such remarkable separation of the chi-rally broken and the deconfined phase might indicate theexistence of a quarkyonic phase, where the quarks arepartly confined to form a Fermi surface, but the relevantdegrees of freedom remain the nucleons with the restoredchiral symmetry [105–109]. Furthermore, in Ref. [110], itwas conjectured that a mixed phase, comprised of nu-cleons and down quarks, may have compelling nontrivialcooling and transport properties that are of interest forthe thermal and rotational evolution of compact stars.

IV. PROPERTIES OF COMPACT STARS

In this section, we explore the physics of dynamicalsequential phase transitions at high baryon density onthe structure of neutron stars.

11

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

8 9 10 11 12 13 14 15 16

M[M

]

R [km]

χ = 0.2χ = 0.1χ = 0

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

8 9 10 11 12 13 14 15 16

PSR J0740+6620

PSR J0030+0451

m0 = 700 MeV

αb0 = 350 MeV

GW170817

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

8 9 10 11 12 13 14 15 16

M[M

]

R [km]

χ = 0.2χ = 0.1χ = 0

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

8 9 10 11 12 13 14 15 16

PSR J0740+6620

PSR J0030+0451

m0 = 700 MeV

αb0 = 370 MeV

GW170817

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

8 9 10 11 12 13 14 15 16

M[M

]

R [km]

χ = 0.2χ = 0.1χ = 0

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

8 9 10 11 12 13 14 15 16

PSR J0740+6620

PSR J0030+0451

m0 = 700 MeV

αb0 = 400 MeV

GW170817

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

8 9 10 11 12 13 14 15 16

M[M

]

R [km]

χ = 0.2χ = 0.1χ = 0

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

8 9 10 11 12 13 14 15 16

PSR J0740+6620

PSR J0030+0451

m0 = 700 MeV

αb0 = 450 MeV

GW170817

FIG. 5. Mass-radius sequences for compact stars as solutions of the TOV equations for m0 = 700 MeV, for αb0 = 350 MeV(upper left), αb0 = 350 MeV (upper right), αb0 = 400 MeV (lower left), αb0 = 450 MeV (lower right). The circles show thecoexistence of the chirally broken and chirally restored phases. The onsets of up and down quarks are marked by pentagonsand diamonds, respectively. We note that, in the upper-left figure, for χ = 0.2, the onset of up and down quarks is almostsimultaneous. The inner (outer) gray band shows the 68.3% (95.4%) credibility regions for the mass of PSR J0740+6620 [36].The inner (outer) green and purple bands show 50% (90%) credibility regions obtained from the recent GW170817 [37] eventfor the low- and high-mass posteriors. The inner (outer) black region corresponds to the mass and radius constraint at 68.2%(95.4%) obtained for PSR J0030+0451 by the group analyzing NICER X-ray data [40].

A. TOV solutions

We used the EoSs introduced in the previous sectionto solve the general-relativistic Tolman-Oppenheimer-Volkoff (TOV) equations [51, 52] for spherically symmet-ric objects,

dP (r)

dr= −

(ε(r) + P (r))(M(r) + 4πr3P (r)

)r (r − 2M(r))

, (31a)

dM(r)

dr= 4πr2ε(r), (31b)

with the boundary conditions P (r = R) = 0,M = M(r = R), where R and M are the radiusand the mass of a neutron star, respectively. Once theinitial conditions are specified based on a given equationof state, namely the central pressure Pc and the central

energy density εc, the internal profile of a neutron starcan be calculated.

In general, there is one-to-one correspondence betweenthe EoS and the mass-radius relation calculated fromEqs. (31). In Fig. 5, we show the relationship of mass ver-sus radius, for the calculated sequences of compact starsfor m0 = 700 MeV, for αb0 = 350 MeV (upper left),αb0 = 370 MeV (upper right), αb0 = 400 MeV (lowerleft), and αb0 = 450 MeV (lower left). In each panel,three calculated sequences for χ = 0 (red: solid line),χ = 0.1 (purple: dashed line), and χ = 0.2 (blue: dot-ted line) are shown. Also shown are the state-of-the-artconstraints: the high-precision mass measurement of thehigh-mass pulsar PSR J0740+6620 [36], constraints fromtwo recent GW170817 [37] events, and the mass-radiusconstraint obtained for PSR J0030+0451 by the groupanalyzing NICER X-ray data [40]. We note that the

12

mass-radius relations obtained in the hybrid QMN modelremain in good agreement with the aforementioned con-straints. In each curve the onsets of down and up quarksare marked with diamonds and pentagons, respectively.

Because the introduction of the quark-vector repulsivecoupling shifts the onset of quarks to higher densities,when compared to the case with vanishing coupling (seeFig. 1), all three sequences in each of the panels share achiral phase transition in the high-mass part (red circle),but still below the 2 M constraint, at around 1.8 M.We note that increasing the value of αb0 softens the chiraltransition, which eventually becomes a smooth crossoverfor αb0 = 450 MeV. We emphasize that an abrupt changein a mass-radius profile in the high-mass part of the se-quence is, in general, a result of a phase transition. Itleads to a softening of the EoS, meaning that it is accom-panied by a rapid flattening of the mass-radius sequence.This is vividly seen in the case of the chiral phase transi-tion on the mass-radius curves in all panels of Fig. 5. Ina model with sequential transitions, a rapid flattening ofthe mass-radius curves is not necessarily associated withthe deconfinement transition, and hence does not dictatethe existence of quark matter in the core of a neutronstar, in contrast to previous studies [111]. Interestingly,the chiral phase transition occurs at masses and radii ac-cessible in the GW190425 merger event [38]. We notethat this is similar to the proton flow constraint fromHIC in the isospin-symmetric matter and the Hebelerconstraint (cf. inset plots in Fig. 1).

The extension of the hadronic branch of an EoS dueto a finite value of the quark vector-interaction χ is alsoreflected in the corresponding mass-radius sequence. In-terestingly, the maximal mass is always reached withinthe hadronic branch of the sequence. For χ = 0, thishappens just before the density jump associated withthe onset of down quark is reached. This is the case,except for αb0 = 370 and 400 MeV, where tiny fractionsof down quarks appear just before the maximal mass isreached (see the top panel of Fig. 2). The appearanceof the down quark makes the matter too soft to sustainfrom the gravitational collapse. For χ = 0.1 and 0.2, thehadronic branch extends beyond the density at which themaximal mass is reached and becomes gravitationally un-stable. This is the case for all values of αb0. Eventually,when down and up quark are sequentially populated, thematter is still not stiff enough to sustain from the collapseand turn into an additional family of stable hybrid com-pact stars. This is shown in Fig. 2, for m0 = 700 MeVand αb0 = 370 MeV, where for different values of χ,the densities at which the maximal mass is reached aremarked by gray solid vertical lines. Clearly, the den-sity for χ = 0.1 (middle panel) is greater than for χ = 0(top panel), and the hadronic branch extends beyond it.For χ = 0.2, the hadronic branch extends even further,however the maximal mass stays the same. Thus, weconclude that a further increase of the quark-vector cou-pling does not support the maximal-mass constraint. InTable V, we show the values of the maximal masses of

a neutron star and corresponding radii obtained in eachparametrization.

We stress that the above structure persists to largeextent when the chiral invariant mass m0 changes. In or-der for the pure quark matter to be present in the coresof neutron stars, the repulsive interactions among themshould be strong. On the other hand, the self-consistenttreatment of the interactions in the hybrid QMN modelalso shifts the onset of the quark matter in the stellar se-quence to higher densities. This eventually prevents theneutron stars with quark matter cores from existence inthe gravitationally stable branch of the sequence. Thisleads to a general conclusion that the existence of hy-brid stars is rather excluded in the current hybrid QMNmodel.

B. Tidal deformability

The dimensionless tidal deformability parameter Λ canbe computed through its relation to the Love numberk2 [112–116],

Λ =2

3k2C

−5, (32)

where C = M/R is the star compactness parameter, withM and R being the total mass and radius of a star. TheLove number k2 reads

k2 =8C5

5(1− 2C)

2[2 + 2C(y − 1)− y]×

×(

2C [6− 3y + 3C(5y − 8)]

+4C3[13− 11y + C(3y − 2) + 2C2(1 + y)]

+3(1− 2C)2[2− y + 2C(y − 1) ln (1− 2C)])−1

,

(33)

where y = Rβ(R)/H(R). The functions H(r) and β(r)are given by the following set of differential equations:

dβ

dr= 2

(1− 2

M(r)

r

)−1

H

−2π

[5ε(r) + 9P (r) +

dε

dP(ε(r) + P (r))

]+

3

r2+ 2

(1− 2

M(r)

r

)−1(M(r)

r2+ 4πrP (r)

)2

+2β

r

(1− 2

M(r)

r

)−1

M(r)

r+ 2πr2(ε(r)− P (r))− 1

, (34)

dH

dr= β. (35)

The above equations have to be solved along withthe TOV equations (31). The initial conditions are

13

0

500

1000

1500

2000

0 250 500 750 1000 1250

Λ2

Λ1

αb0 = 350 MeVαb0 = 370 MeVαb0 = 400 MeVαb0 = 450 MeV

0

500

1000

1500

2000

0 250 500 750 1000 1250

50%

90%

FIG. 6. Relation between tidal deformabilities Λ1 and Λ2 oftwo compact stars that merged in the GW170817 event [37].For comparison, also shown are the 50% and 90% fidelityregions from the analysis of the GW signal by the LIGO-VIRGO collaboration [37]. The gray shading corresponds tothe unphysical region Λ2 < Λ1.

H(r → 0) = c0r2 and β(r → 0) = 2c0r, where c0 is a con-

stant, which is irrelevant in the expression for the Lovenumber k2.

In Fig. 6, we plot the tidal deformability parametersΛ1 vs λ2 of the high- and low-mass members of the bi-nary merger together with the 50% and 90% fidelity re-gions obtained by the LVC analysis of the GW170817event [37]. Because the extracted most probable massesof the members are below the mass where the chiral phasetransition occurs (cf. Fig. 5), the Λ1 vs. Λ2 diagrams arethe same for given m0 and αb0, irrespective of the valueof the parameter χ. We note that the tidal deformabil-ity parameter requires sufficiently soft EoS around thesaturation density. This is seen in the figure, where thesmallest tidal deformability and thus the best agreementwith the constraint is obtained for the highest value ofαb0 = 450 MeV, which corresponds to the softest EoSat low density. We remark that this is in similar fashionto the proton flow constraint. On the other hand, the2 M requires a sufficiently stiff EoS at higher densities.In Table V, we show the maximal masses obtained foreach set of parameters. Inversely to the tidal deforma-bility, the most massive stars are obtained for the stiffestEoS, namely for αb0 = 350 MeV. In general, the twoconstraints are of exclusive character. The interplay be-tween them could be further used to fix the allowed rangeof external model parameters. This would be of partic-ular use in a class of effective models in which the low-and high-density regimes are not treated independently,but rather combined in a consistent unified framework.

V. CONCLUSIONS

Despite the success of LQCD at a finite temperatureand vanishing density, the nature of the chiral and de-confinement phase transitions at a low temperature andhigh density remains unknown. At present, the investiga-tion of matter under such extreme conditions is almostexclusively considered within either purely hadronic orpurely quark effective models, with only few approachesto bridge the gap between them. They are typically basedon excluded volume approach or a Polyakov loop repre-senting a phenomenological description of the transitionfrom the confined to the deconfined phase [24, 117–120].

In this work, we utilized a superior approach to anyprevious modeling, the hybrid QMN model [16], to quan-tify the EoS of cold and dense nuclear matter. This modelgoes beyond the common two-phase approach to the EoSand embeds the interplay between the quark confinementand the chiral symmetry breaking in a dynamical wayinto a single unified framework. We have extended theprevious version of the model by including repulsive in-teractions among quarks mediated by the exchange ofvector mesons. Within this approach, we systematicallyinvestigated the EoS of cold and dense isospin-symmetricmatter, as well as asymmetric matter under NS condi-tions. We constructed the mass-radius relations basedon solutions of the TOV equations.

We find that the class of EoSs obtained in the consid-ered model feature sequential chiral and deconfinementphase transitions. We demonstrate that, as a conse-quence of the realization of the chiral symmetry restora-tion by parity doubling within the hadronic phase, astrong first-order phase transition invalidates the impli-cation that a flattening, eventually even occurrence ofa mass-twin phenomenon, of the mass-radius relation forcompact stars at 2 M, could inevitably signal the decon-finement phase transition in compact stars [111]. We ver-ified that such flattening appears due to the chiral phasetransition which also lies within the region of the protonflow constraint in isospin-symmetric matter [95], as wellas within the constraint derived in Ref. [42] for the mat-ter under NS conditions. Moreover, the chiral transitionis featured in the mass region accessible by the recentGW190425 merger event [38], that is, 1.46 − 1.87 M,which is of particular interest from the observational per-spectives.

The EoS obtained in the hybrid QMN model, underthe NS conditions, is comprised of four consecutive den-sity regions, from low to high density: confined and chi-rally broken phase, confined and chirally restored phase,partially deconfined, and deconfined phase. In the in-termediate phases, the confined and deconfined mattermay coexist as what could be a possible manifestation ofthe conjectured quarkyonic phase [105–109]. We stressthat such a sequential structure of an EoS cannot beobtained in a class of pure quark models or even in aclass of hadron-quark models, where the confined and de-confined phases are treated independently. Recent stud-

14

ies in this direction provide an interesting scenario for arapid increase in the pressure within densities achievedin NSs [108, 109]. It would also be interesting to estab-lish the cooling and transport properties of the hybridQMN model EoS, especially the mixed phase comprisedof nucleons and down quarks, as well as to establish theirinfluence on the thermal and rotational evolution of com-pact objects.

We embedded the repulsive quark-vector interactionsin the QMN model and studied their consequences forthe phenomenological description of compact stellar ob-jects. In particular, we analyzed a possible occurrenceof a quark matter in the NS core. The inclusion ofthe quark-vector repulsion implies a shift of the onsetof quarks to higher densities. This results in the mass-radius diagram with the maximal mass around 2 M,which is within the chirally restored hadronic configura-tion.

In the present studies, the transition to pure quarkmatter is likely to appear in the part of the stellar se-quence that is already gravitationally unstable. We note,however, that the presence of the attractive diquark in-teraction can induce scalar diquark condensation in thequark phase and reduce the density of the onset of thequark matter [53]. This comes with a softening of the EoSwhich, however, can be compensated by a stronger vec-tor meson coupling [54, 121]. Moreover, in recent LQCDstudies, it was found that parity doubling also occursin the hyperon channels [11]. The inclusion of heavierflavors is known to soften the equation of state. There-fore, it would be of further interest to consider the QMNequations of state that include the attractive diquark in-teraction and the hyperon degrees of freedom in orderto test a model’s compliance with the 2 M constraint.Work in this direction is already in progress.

We show how modern astrophysical constraints onthe maximum-mass [36], the tidal deformability fromthe binary merger GW170817 [37], and recent simulta-neous mass-radius constraints from the NICER experi-ment [39, 40], compiled together, allow for a consistentdetermination of parameters characterizing the EoS ofdense nuclear matter. We demonstrate that not only toosoft (excluded by the maximum mass constraint) but alsotoo stiff (the tidal deformability constraint) equations ofstate may be ruled out in the current approach. Withthe optimal choice of parameters in the QMN model, we

introduced a class of EoSs that fulfill the above astro-physical constraints, as well as phenomenological con-straints imposed by flow data from heavy ion collisions.This makes the present hybrid QMN model EoSs su-perior to previously proposed class of EoSs obtained inpure-hadronic or pure-quark models, as well as in hybridmodels based on a two-phase approach. It would be ofinterest to apply a Bayesian analysis method for selectingthe most probable equation of state under a set of mod-ern constraints from compact star physics, and from dataobtained in heavy ion collisions. For applications of theQMN model EoS to multi-messenger astronomy in simu-lations of merger or supernova events as well as heavy ioncollision, it is necessary to perform a finite-temperatureextension. This is a straightforward task in the presentedfield-theoretical model as first results for correspondingphase diagrams of the unified quark-hadron matter de-scription convincingly demonstrate [18, 19].

We provided a first step toward a unified quark-hadronmatter EoS for multi-messenger astronomy that allowsconclusions for the composition of neutron star interiorsand the phase structure of matter under extreme con-ditions, in accordance with mass-radius measurementsof compact stars and heavy ion collision phenomenol-ogy. The next steps include be the generalization of theQMN model to include color superconducting quark mat-ter phases that may allow the existence of hybrid quark-hadron stars due to the lowering of the onset mass fordeconfinement while fulfilling the maximum mass con-straint for a sufficiently large coupling to vector mesonfields.

ACKNOWLEDGMENTS

This work was partly supported by the Polish Na-tional Science Center (NCN), under OPUS Grant No.2018/31/B/ST2/01663 (K.R. and C.S.), Opus Grant No.UMO-2019/33/B/ST9/03059 (D.B.), and PreludiumGrant No. UMO-2017/27/N/ST2/01973 (M.M.). D.B.received support from the National Research NuclearUniversity (MEPhI) within the Russian Academic Excel-lence Project under contract No. 02.a03.21.0005. K.R.also acknowledges stimulating discussions with LarryMcLerran and the support of the Polish Ministry of Sci-ence and Higher Education.

[1] Y. Aoki, G. Endrodi, Z. Fodor, S. Katz, and K. Szabo,Nature 443, 675 (2006), arXiv:hep-lat/0611014.

[2] A. Bazavov et al. (HotQCD), Phys. Lett. B 795, 15(2019), arXiv:1812.08235 [hep-lat].

[3] S. Ejiri, F. Karsch, and K. Redlich, Phys. Lett. B 633,275 (2006), arXiv:hep-ph/0509051.

[4] F. Karsch and K. Redlich, Phys. Lett. B 695, 136(2011), arXiv:1007.2581 [hep-ph].

[5] S. Borsanyi, Z. Fodor, S. D. Katz, S. Krieg, C. Ratti,

and K. Szabo, JHEP 01, 138 (2012), arXiv:1112.4416[hep-lat].

[6] A. Bazavov et al. (HotQCD), Phys. Rev. D86, 034509(2012), arXiv:1203.0784 [hep-lat].

[7] A. Bazavov et al. (HotQCD), Phys. Rev. D90, 094503(2014), arXiv:1407.6387 [hep-lat].

[8] A. Bazavov et al., (2020), arXiv:2001.08530 [hep-lat].[9] G. Aarts, C. Allton, S. Hands, B. Jger, C. Praki,

and J.-I. Skullerud, Phys. Rev. D92, 014503 (2015),

15

arXiv:1502.03603 [hep-lat].[10] G. Aarts, C. Allton, D. De Boni, S. Hands, B. Jger,

C. Praki, and J.-I. Skullerud, JHEP 06, 034 (2017),arXiv:1703.09246 [hep-lat].

[11] G. Aarts, C. Allton, D. De Boni, and B. Jger, Phys.Rev. D99, 074503 (2019), arXiv:1812.07393 [hep-lat].

[12] C. E. De Tar and T. Kunihiro, Phys. Rev. D39, 2805(1989).

[13] D. Jido, T. Hatsuda, and T. Kunihiro, Phys. Rev. Lett.84, 3252 (2000), arXiv:hep-ph/9910375 [hep-ph].

[14] D. Jido, M. Oka, and A. Hosaka, Prog. Theor. Phys.106, 873 (2001), arXiv:hep-ph/0110005 [hep-ph].

[15] D. Zschiesche, L. Tolos, J. Schaffner-Bielich, and R. D.Pisarski, Phys. Rev. C75, 055202 (2007), arXiv:nucl-th/0608044 [nucl-th].

[16] S. Benic, I. Mishustin, and C. Sasaki, Phys. Rev. D91,125034 (2015), arXiv:1502.05969 [hep-ph].

[17] M. Marczenko and C. Sasaki, Phys. Rev. D97, 036011(2018), arXiv:1711.05521 [hep-ph].

[18] M. Marczenko, D. Blaschke, K. Redlich, and C. Sasaki,Phys. Rev. D98, 103021 (2018), arXiv:1805.06886.

[19] M. Marczenko, D. Blaschke, K. Redlich, and C. Sasaki,Universe 5, 180 (2019), arXiv:1905.04974 [nucl-th].

[20] A. Motornenko, J. Steinheimer, V. Vovchenko,S. Schramm, and H. Stoecker, Phys. Rev. C101, 034904(2020), arXiv:1905.00866 [hep-ph].

[21] A. Mukherjee, S. Schramm, J. Steinheimer, andV. Dexheimer, Astron. Astrophys. 608, A110 (2017),arXiv:1706.09191 [nucl-th].

[22] A. Mukherjee, J. Steinheimer, and S. Schramm, Phys.Rev. C96, 025205 (2017), arXiv:1611.10144 [nucl-th].

[23] V. Dexheimer, J. Steinheimer, R. Negreiros, andS. Schramm, Phys. Rev. C87, 015804 (2013),arXiv:1206.3086 [astro-ph.HE].

[24] J. Steinheimer, S. Schramm, and H. Stocker, Phys. Rev.C84, 045208 (2011), arXiv:1108.2596 [hep-ph].

[25] J. Weyrich, N. Strodthoff, and L. von Smekal, Phys.Rev. C92, 015214 (2015), arXiv:1504.02697 [nucl-th].

[26] C. Sasaki and I. Mishustin, Phys. Rev. C82, 035204(2010), arXiv:1005.4811 [hep-ph].

[27] T. Yamazaki and M. Harada, Phys. Rev. C100, 025205(2019), arXiv:1901.02167 [nucl-th].

[28] T. Ishikawa, K. Nakayama, and K. Suzuki, Phys. Rev.D99, 054010 (2019), arXiv:1812.10964 [hep-ph].

[29] J. Steinheimer, S. Schramm, and H. Stocker, J. Phys.G38, 035001 (2011), arXiv:1009.5239 [hep-ph].

[30] A. Motornenko, V. Vovchenko, J. Steinheimer,S. Schramm, and H. Stoecker, Proceedings, 27thInternational Conference on Ultrarelativistic Nucleus-Nucleus Collisions (Quark Matter 2018): Venice, Italy,May 14-19, 2018, Nucl. Phys. A982, 891 (2019),arXiv:1809.02000 [hep-ph].

[31] A. Bauswein, N.-U. F. Bastian, D. B. Blaschke,K. Chatziioannou, J. A. Clark, T. Fischer, andM. Oertel, Phys. Rev. Lett. 122, 061102 (2019),arXiv:1809.01116 [astro-ph.HE].

[32] T. Fischer, N.-U. F. Bastian, M.-R. Wu, P. Baklanov,E. Sorokina, S. Blinnikov, S. Typel, T. Klahn, and D. B.Blaschke, Nat. Astron. 2, 980 (2018), arXiv:1712.08788[astro-ph.HE].

[33] P. Demorest, T. Pennucci, S. Ransom, M. Roberts, andJ. Hessels, Nature 467, 1081 (2010), arXiv:1010.5788[astro-ph.HE].

[34] J. Antoniadis et al., Science 340, 6131 (2013),

arXiv:1304.6875 [astro-ph.HE].[35] E. Fonseca et al., Astrophys. J. 832, 167 (2016),

arXiv:1603.00545 [astro-ph.HE].[36] H. T. Cromartie et al., Nat. Astron. 4, 72 (2019),

arXiv:1904.06759 [astro-ph.HE].[37] B. P. Abbott et al. (LIGO Scientific, Virgo), Phys. Rev.

Lett. 121, 161101 (2018), arXiv:1805.11581 [gr-qc].[38] B. P. Abbott et al. (LIGO Scientific, Virgo), (2020),

arXiv:2001.01761 [astro-ph.HE].[39] T. E. Riley et al., Astrophys. J. 887, L21 (2019),

arXiv:1912.05702 [astro-ph.HE].[40] M. C. Miller et al., Astrophys. J. 887, L24 (2019),

arXiv:1912.05705 [astro-ph.HE].[41] M. G. Alford, S. Han, and M. Prakash, Phys. Rev.

D88, 083013 (2013), arXiv:1302.4732 [astro-ph.SR].[42] K. Hebeler, J. M. Lattimer, C. J. Pethick,

and A. Schwenk, Astrophys. J. 773, 11 (2013),arXiv:1303.4662 [astro-ph.SR].

[43] J. S. Read, B. D. Lackey, B. J. Owen, and J. L. Fried-man, Phys. Rev. D79, 124032 (2009), arXiv:0812.2163[astro-ph].

[44] D. E. Alvarez-Castillo and D. B. Blaschke, Phys. Rev.C96, 045809 (2017), arXiv:1703.02681 [nucl-th].

[45] Blaschke, David, Schaffner-Bielich, Jurgen, andSchulze, Hans-Josef, Eur. Phys. J. A 52, 71 (2016).

[46] N. K. Glendenning, “General Relativity and CompactStars,” in Centennial of General Relativity: A Celebra-tion, edited by C. A. Zen Vasconcellos (2017) pp. 1–72.

[47] M. Oertel, M. Hempel, T. Klahn, and S. Typel, Rev.Mod. Phys. 89, 015007 (2017), arXiv:1610.03361 [astro-ph.HE].

[48] I. Bombaci, JPS Conf. Proc. 17, 101002 (2017),arXiv:1601.05339 [nucl-th].

[49] M. Baldo, G. F. Burgio, and H.-J. Schulze, in Su-perdense QCD matter and compact stars. Proceedings,NATO Advanced Research Workshop, Erevan, Arme-nia, September 27-October 4, 2003 (2003) pp. 113–134,arXiv:astro-ph/0312446.

[50] M. Shahrbaf, D. Blaschke, A. Grunfeld, andH. Moshfegh, Phys. Rev. C 101, 025807 (2020),arXiv:1908.04740 [nucl-th].

[51] R. C. Tolman, Phys. Rev. 55, 364 (1939).[52] J. R. Oppenheimer and G. M. Volkoff, Phys. Rev. 55,

374 (1939).[53] T. Klahn, D. Blaschke, F. Sandin, C. Fuchs, A. Faessler,

H. Grigorian, G. Ropke, and J. Trumper, Phys. Lett.B654, 170 (2007), arXiv:nucl-th/0609067 [nucl-th].

[54] T. Klahn, R. astowiecki, and D. Blaschke, Phys.Rev.D88, 085001 (2013), arXiv:1307.6996 [nucl-th].

[55] S. Benic, D. Blaschke, D. E. Alvarez-Castillo, T. Fis-cher, and S. Typel, Astron. Astrophys. 577, A40(2015), arXiv:1411.2856 [astro-ph.HE].

[56] T. Klahn and T. Fischer, Astrophys. J. 810, 134 (2015),arXiv:1503.07442 [nucl-th].

[57] M. Kitazawa, T. Koide, T. Kunihiro, and Y. Nemoto,Prog. Theor. Phys. 108, 929 (2002), [Erratum:Prog. Theor. Phys.110,no.1,185(2003)], arXiv:hep-ph/0207255 [hep-ph].

[58] J. E. Trumper, V. Burwitz, F. Haberl, and V. E. Za-vlin, The restless high-energy universe. Proceedings, 2ndBeppoSAX Conference, BeppoSAX 2003, Amsterdam,Netherlands, May 5-8, 2003, Nucl. Phys. Proc. Suppl.132, 560 (2004), [,560(2003)], arXiv:astro-ph/0312600[astro-ph].

16

[59] F. zel, Nature 441, 1115 (2006), arXiv:astro-ph/0605106 [astro-ph].

[60] M. Alford, D. Blaschke, A. Drago, T. Klahn,G. Pagliara, and J. Schaffner-Bielich, Nature 445, E7(2007), arXiv:astro-ph/0606524 [astro-ph].

[61] K. Masuda, T. Hatsuda, and T. Takatsuka, Astrophys.J. 764, 12 (2013), arXiv:1205.3621 [nucl-th].

[62] C. H. Lenzi and G. Lugones, Astrophys. J. 759, 57(2012), arXiv:1206.4108 [astro-ph.SR].

[63] M. Orsaria, H. Rodrigues, F. Weber, and G. A. Con-trera, Phys. Rev. D87, 023001 (2013), arXiv:1212.4213[astro-ph.SR].

[64] M. Orsaria, H. Rodrigues, F. Weber, and G. A. Con-trera, Phys. Rev. C89, 015806 (2014), arXiv:1308.1657[nucl-th].

[65] M. Fortin, A. R. Raduta, S. Avancini, and C. Provid-ncia, (2020), arXiv:2001.08036 [hep-ph].

[66] N.-U. Bastian and D. Blaschke, Proceedings, 15th In-ternational Conference on Strangeness in Quark Mat-ter (SQM 2015): Dubna, Moscow region, Russia, July6-11, 2015, J. Phys. Conf. Ser. 668, 012042 (2016),arXiv:1511.05881 [nucl-th].

[67] K. Fukushima, Phys. Lett. B591, 277 (2004),arXiv:hep-ph/0310121 [hep-ph].

[68] C. Ratti, M. A. Thaler, and W. Weise, Phys. Rev. D73,014019 (2006), arXiv:hep-ph/0506234 [hep-ph].

[69] C. Sasaki, B. Friman, and K. Redlich, Phys. Rev. D75,074013 (2007), arXiv:hep-ph/0611147 [hep-ph].

[70] S. Roessner, C. Ratti, and W. Weise, Phys. Rev. D75,034007 (2007), arXiv:hep-ph/0609281 [hep-ph].

[71] K. Fukushima, Phys. Rev. D77, 114028 (2008), [Er-ratum: Phys. Rev.D78,039902(2008)], arXiv:0803.3318[hep-ph].

[72] B.-J. Schaefer, J. M. Pawlowski, and J. Wambach,Phys. Rev. D76, 074023 (2007), arXiv:0704.3234 [hep-ph].

[73] V. Skokov, B. Stokic, B. Friman, and K. Redlich, Phys.Rev. C82, 015206 (2010), arXiv:1004.2665 [hep-ph].

[74] V. Skokov, B. Friman, and K. Redlich, Phys. Rev. C83,054904 (2011), arXiv:1008.4570 [hep-ph].

[75] A. J. Mizher, M. N. Chernodub, and E. S. Fraga, Phys.Rev. D82, 105016 (2010), arXiv:1004.2712 [hep-ph].

[76] T. K. Herbst, J. M. Pawlowski, and B.-J. Schaefer,Phys. Lett. B696, 58 (2011), arXiv:1008.0081 [hep-ph].

[77] T. Fischer, T. Klahn, and M. Hempel, Eur. Phys. J.A52, 225 (2016), arXiv:1608.04613 [nucl-th].

[78] T. Klahn, T. Fischer, and M. Hempel, Astrophys. J.836, 89 (2017), arXiv:1603.03679 [nucl-th].

[79] A. Bauswein, N.-U. F. Bastian, D. Blaschke, K. Chatzi-ioannou, J. A. Clark, T. Fischer, H.-T. Janka, O. Just,M. Oertel, and N. Stergioulas, Proceedings, Xiamen-CUSTIPEN Workshop on the Equation of DenseNeutron-Rich Matter in the Era of Gravitational WaveAstronomy: Xiamen, China, January 3-7, 2019, AIPConf. Proc. 2127, 020013 (2019), arXiv:1904.01306[astro-ph.HE].

[80] T. Fischer, M.-R. Wu, B. Wehmeyer, N.-U. F. Bastian,G. Martnez-Pinedo, and F.-K. Thielemann, (2020),arXiv:2003.00972 [astro-ph.HE].

[81] J. P. Pereira, M. Bejger, N. Andersson, and F. Gittins,(2020), arXiv:2003.10781 [gr-qc].

[82] C. Sasaki, Phys. Lett. B801, 135172 (2020),arXiv:1906.05077 [hep-ph].

[83] J. D. Walecka, Annals Phys. 83, 491 (1974).

[84] O. Scavenius, A. Mocsy, I. N. Mishustin, and D. H.Rischke, Phys. Rev. C64, 045202 (2001), arXiv:nucl-th/0007030 [nucl-th].

[85] S. Shlomo, V. M. Kolomietz, and G. Col, Eur. Phys. JA30, 23 (2006).

[86] Y. Motohiro, Y. Kim, and M. Harada, Phys.Rev. C92, 025201 (2015), [Erratum: Phys.Rev.C95,no.5,059903(2017)], arXiv:1505.00988 [nucl-th].

[87] M. Tanabashi et al. (Particle Data Group), Phys. Rev.D98, 030001 (2018).

[88] J. I. Kapusta and C. Gale, Finite-temperature fieldtheory: Principles and applications, Cambridge Mono-graphs on Mathematical Physics (Cambridge UniversityPress, 2011).

[89] M. A. R. Kaltenborn, N.-U. F. Bastian, andD. B. Blaschke, Phys. Rev. D96, 056024 (2017),arXiv:1701.04400 [astro-ph.HE].

[90] M. G. Alford, G. F. Burgio, S. Han, G. Taranto,and D. Zappal, Phys. Rev. D92, 083002 (2015),arXiv:1501.07902 [nucl-th].

[91] H. Roberts, C. Roberts, A. Bashir, L. Gutierrez-Guerrero, and P. Tandy, Phys. Rev. C 82, 065202(2010), arXiv:1009.0067 [nucl-th].

[92] H. Roberts, A. Bashir, L. Gutierrez-Guerrero,C. Roberts, and D. Wilson, Phys. Rev. C 83,065206 (2011), arXiv:1102.4376 [nucl-th].

[93] A. Cherman, S. Sen, M. Unsal, M. L. Wagman, andL. G. Yaffe, Phys. Rev. Lett. 119, 222001 (2017),arXiv:1706.05385 [hep-th].

[94] K. Aitken, A. Cherman, E. Poppitz, and L. G. Yaffe,Phys. Rev. D 96, 096022 (2017), arXiv:1707.08971 [hep-th].

[95] P. Danielewicz, R. Lacey, and W. G. Lynch, Science298, 1592 (2002), arXiv:nucl-th/0208016 [nucl-th].

[96] A. Andronic, P. Braun-Munzinger, J. Stachel, andM. Winn, Phys. Lett. B 718, 80 (2012), arXiv:1201.0693[nucl-th].

[97] D. Blaschke, H. Grigorian, and G. Rpke, Particles 3,477 (2020), arXiv:2005.10218 [nucl-th].

[98] T. Krger, I. Tews, K. Hebeler, and A. Schwenk, Phys.Rev. C 88, 025802 (2013), arXiv:1304.2212 [nucl-th].

[99] B.-A. Li and X. Han, Phys. Lett. B 727, 276 (2013),arXiv:1304.3368 [nucl-th].

[100] S. Typel, Eur. Phys. J. A52, 16 (2016).[101] M. G. Alford and A. Sedrakian, Phys. Rev. Lett. 119,

161104 (2017), arXiv:1706.01592 [astro-ph.HE].[102] V. Paschalidis, K. Yagi, D. Alvarez-Castillo, D. B.

Blaschke, and A. Sedrakian, Phys. Rev. D97, 084038(2018), arXiv:1712.00451 [astro-ph.HE].

[103] C. Sasaki, B. Friman, and K. Redlich, Phys. Rev. D75,054026 (2007), arXiv:hep-ph/0611143 [hep-ph].

[104] N. M. Bratovic, T. Hatsuda, and W. Weise, Phys. Lett.B719, 131 (2013), arXiv:1204.3788 [hep-ph].

[105] Y. Hidaka, L. D. McLerran, and R. D. Pisarski, Nucl.Phys. A808, 117 (2008), arXiv:0803.0279 [hep-ph].

[106] L. McLerran, K. Redlich, and C. Sasaki, Nucl. Phys.A824, 86 (2009), arXiv:0812.3585 [hep-ph].

[107] A. Andronic et al., Nucl. Phys. A837, 65 (2010),arXiv:0911.4806 [hep-ph].

[108] L. McLerran and S. Reddy, Phys. Rev. Lett. 122,122701 (2019), arXiv:1811.12503 [nucl-th].

[109] K. S. Jeong, L. McLerran, and S. Sen, (2019),arXiv:1908.04799 [nucl-th].

17

[110] D. Blaschke, F. Sandin, T. Klahn, and J. Berdermann,Phys. Rev. C80, 065807 (2009), arXiv:0807.0414 [nucl-th].

[111] D. Alvarez-Castillo, S. Benic, D. Blaschke, S. Han,and S. Typel, Eur. Phys. J. A52, 232 (2016),arXiv:1608.02425 [nucl-th].

[112] T. Hinderer, Astrophys. J. 677, 1216 (2008),arXiv:0711.2420 [astro-ph].

[113] T. Damour and A. Nagar, Phys. Rev. D80, 084035(2009), arXiv:0906.0096 [gr-qc].

[114] T. Binnington and E. Poisson, Phys. Rev. D80, 084018(2009), arXiv:0906.1366 [gr-qc].

[115] K. Yagi and N. Yunes, Phys. Rev. D88, 023009 (2013),arXiv:1303.1528 [gr-qc].

[116] T. Hinderer, B. D. Lackey, R. N. Lang, and J. S. Read,Phys. Rev. D81, 123016 (2010), arXiv:0911.3535 [astro-ph.HE].

[117] J. Berges, D. U. Jungnickel, and C. Wetterich, Int. J.Mod. Phys. A18, 3189 (2003), arXiv:hep-ph/9811387[hep-ph].

[118] J. Meyer, K. Schwenzer, and H. J. Pirner, Phys. Lett.B473, 25 (2000), arXiv:nucl-th/9908017 [nucl-th].

[119] S. Lawley, W. Bentz, and A. W. Thomas, J. Phys. G32,667 (2006), arXiv:nucl-th/0602014 [nucl-th].

[120] V. A. Dexheimer and S. Schramm, Phys. Rev. C81,045201 (2010), arXiv:0901.1748 [astro-ph.SR].

[121] D. Alvarez-Castillo, D. Blaschke, A. Grunfeld, andV. Pagura, Phys. Rev. D 99, 063010 (2019),arXiv:1805.04105 [hep-ph].

Appendix A: Impact of statistical confinement onequation of state

In this appendix, we provide an example of the stiffen-ing of the EoS due to the statistical confinement in thehybrid QMN model. For illustration, let us consider twoEoSs. For the first EoS, we take a non-interacting gasof massless fermions at zero temperature. In this case,the pressure, P1(µ), can be evaluated analytically as afunction of chemical potential µ, as

P1(µ) = −µ∫

0

dp

2π2p2 (p− µ) =

µ4

24π2, (A1)

where the upper integration limit is due to the Fermi-Dirac distribution function for a massless particle in thezero-temperature limit, θ (µ− p). For the second EoS, letus consider the modification of the distribution functionas in Eq. (10). For simplicity, we also set α = 1. Thepressure, P2(µ), is then obtained as

P2(µ) = −µ∫

0

dp

2π2p2 (p− µ) θ (b− p) =

µ4

24π2 , if b ≥ µb4

24π2 , if b < µ.

(A2)Thus, if b ≥ µ, the pressure P2(µ) evaluates to P1(µ).However, if b < µ, the mechanism of statistical confine-ment sets in and the integration region is limited up tob, and pressure P2(µ) < P1(µ). Let us assume that inthe region where b < µ, the functional form of the bfield is b = γµ, where γ is a positive constant, such thatγ ∈ (0, 1). Then we get the following:

P2(µ) =b4

24π2= γ4

µ4

24π2= γ4P1(µ) < P1(µ). (A3)

Let us consider two values of chemical potential, µ1 ¡ µ2,such that P1(µ1) = P2(µ2). From this, it follows thatµ41 = γ4µ4

2. Because the density is defined as the deriva-tive of pressure w.r.t chemical potential, ρ = ∂P/∂µ, weget

ρ2(µ2) = γ4µ32

6π2=µ1

µ2

µ31

6π2=µ1

µ2ρ1(µ1) < ρ1(µ1). (A4)

The last inequality holds by the assumption that µ1 <µ2. Thus, the pressure P2 increases faster than P1 as afunction of density ρ. This is the stiffening effect observedin the class of EoSs obtained in the hybrid QMN model.We note that, conversely to the case discussed in thisappendix, the appearance of quarks always provides anextra contribution to the pressure, thus, softening thecorresponding EoS.